Relieving the post-selection problem by quantum singular value transformation

Quantum measurement is a fundamental yet experimentally challenging ingredient of quantum information processing. Many recent studies on quantum dynamics focus on expectation values of nonlinear observables; however, their experimental measurement is hindered by the post-selection problem -- namely, the substantial overhead caused by uncontrollable measurement outcomes. In this work, we propose a post-selection--free experimental strategy based on a fully quantum approach. The key idea is to deterministically simulate the post-selected quantum states by applying quantum singular value transformation (QSVT) algorithms. For pure initial state post-selection, our method is a generalization of fixed-point amplitude amplification to arbitrary projective measurements, achieving an optimal quadratic speedup. We further extend this framework to mixed initial state post-selection by applying linear amplitude amplification via QSVT, which significantly enhances the measurement success probability. However, a deterministic quantum algorithm for preparing the post-selected mixed state is generally impossible because of information-theoretic constraints imposed by quantum coding theory. Additionally, we introduce a pseudoinverse decoder for measurement-induced quantum teleportation. This decoder possesses the novel property that, when conditioned on a successful flag measurement, the decoding is nearly perfect even in cases where channel decoders are information-theoretically impossible. Overall, our work establishes a powerful approach for measuring novel quantum dynamical phenomena and presents quantum algorithms as a new perspective for understanding quantum dynamics and quantum chaos.